I wouldn't do it unless they won't fit in the box for some reason. I haven't packed a bike in awhile, but it seems like all I did was take off the pedals, loosen the headset to turn the bars, take off the seat with post, and remove the wheels. Sometimes you can just remove the front wheel. Oh, and put a spacer where the wheels went. so the dropouts don't get squished.

Note: with the lowest spacer on a threadless style fork being a clamp
one like the seat tube to hold the seat post or there is a thin one
from QBP they call a 'locking spacer', you can loosen the Top cap
and just remove the stem.

Quill stems are always easy to pull out, though slotted cable stops make

It easy to take the cables out to not have to re adjust them after re installing the bars ..

I got the suggestion at one airport , I should have Busked the reassembly show

with a tip jar,
at the arrival carrousel as it may have earned me a bit of the local currency
in exchange for the entertainment.

how about loosening the stem faceplate and rotating the bars such that the brake handles are pointed downward?

When I shipped my bike, I would use a bike box from the bike shop. To fit it into the box it required the wheels, handlebars, pedals and seatpost to be removed. I would also put supports in the fork end and dropouts, so the fork and frame would not be bent, an old pair of hubs would work also.